Inverse problems for elliptic equations with power type nonlinearities
| Autor: | Yi-Hsuan Lin, Tony Liimatainen, Matti Lassas, Mikko Salo |
|---|---|
| Přispěvatelé: | Department of Mathematics and Statistics, Inverse Problems, Matti Lassas / Principal Investigator |
| Rok vydání: | 2021 |
| Předmět: |
Mathematics - Differential Geometry
GLOBAL UNIQUENESS General Mathematics Conformal map CALDERON PROBLEM Transversally anisotropic 01 natural sciences inversio-ongelmat Mathematics - Analysis of PDEs Simple (abstract algebra) Euclidean geometry FOS: Mathematics 111 Mathematics Applied mathematics 0101 mathematics Mathematics Inverse boundary value problem osittaisdifferentiaaliyhtälöt Calderón problem Geometrical optics Semilinear equation Applied Mathematics 010102 general mathematics transversally anisotropic Inverse problem Manifold 010101 applied mathematics semilinear equation Nonlinear system Differential Geometry (math.DG) inverse boundary value problem Linear equation Analysis of PDEs (math.AP) |
| Zdroj: | Journal de Mathématiques Pures et Appliquées |
| ISSN: | 0021-7824 |
| DOI: | 10.1016/j.matpur.2020.11.006 |
| Popis: | We introduce a method for solving Calder\'on type inverse problems for semilinear equations with power type nonlinearities. The method is based on higher order linearizations, and it allows one to solve inverse problems for certain nonlinear equations in cases where the solution for a corresponding linear equation is not known. Assuming the knowledge of a nonlinear Dirichlet-to-Neumann map, we determine both a potential and a conformal manifold simultaneously in dimension $2$, and a potential on transversally anisotropic manifolds in dimensions $n \geq 3$. In the Euclidean case, we show that one can solve the Calder\'on problem for certain semilinear equations in a surprisingly simple way without using complex geometrical optics solutions. Comment: 25 pages |
| Databáze: | OpenAIRE |
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